Lissajous Curves
For the parametric equations x=4sin([a/b]t) and y=3sin(t) where 0<t<30 and a/b=1/2, 1/4, 2/1, 4/1, and 2/3 we can see that the curves x=4sin(at) and y=3sin(bt) gives the same graph.
This pattern does not hold true when a/b=12/13 though as shown below:
In fact, the graphs are different when a/b=4/5. Hence I conjecture that the graphs differ when a is even and b is odd and a<b (and when a/b>2/3)
Notice that for both the graphs where this does not hold true, at least one of the graphs is not closed.
Looking at the graphs for x=4sin([a/b]t) and y=sin(t), we can see that the values of a and b will predict what the graph will look like. The value for a is the number of times that the graph touches the left or right edge of the range. The value for b is the number of maximums or minimums in the graphs.
In general this will work for the second set of equations, x=4sin(at) and y=sin(bt). If a or b is odd, this will cause the graph to not be closed, so these values not hold true for this set of equations. In fact, if the values for a and b become large enough, the graphs of the different sets of equations begin to look nothing like each other (see above).
